A TREATISE ON DYNAMICS - MECHANICAL PROBLEMS
CONTENTS CHAPTER 1 INTRODUCTION TO INVERSE METHODS 1 1 INTRODUCTION 1 2 IDENTIFICATION METHODS 1 3 IDENTIFICATION OF THE STRAIN HARDENING LAW 1 3 1 EXAMPLE OF AN APPLICATION 1 3 2 VALIDATION TEST 1 3 3 HYDROFORMING A WELDED TUBE CHAPTER 2 LINEAR DIFFERENTIAL EQUATION SYSTEMS OF THE FIRST ORDER WITH CONSTANT COEFFICIENTS: APPLICATION IN MECHANICAL ENGINEERING 2 1 INTRODUCTION 2 2 MODELING DISSIPATIVE SYSTEMS 2 2 1 INTRINSIC SOLUTIONS OF AUTONOMOUS SYSTEMS 2 2 2 INTRINSIC SOLUTIONS 2 2 3 INTRINSIC SOLUTIONS OF THE ADJOINING SYSTEM 2 2 4 RELATION BETWEEN THE INTRINSIC SOLUTIONS OF S AND S* 2 2 5 RELATION BETWEEN MODAL MATRICES X AND X* 2 3 AUTONOMOUS SYSTEM GENERAL SOLUTION 2 3 1 DIRECT SOLUTION BY USING THE EXPONENTIAL MATRIX 2 3 2 INDIRECT SOLUTION BY MODAL TRANSFORMATION 2 4 GENERAL SOLUTION OF THE COMPLETE EQUATION 2 4 1 DIRECT SOLUTION BY THE EXPONENTIAL MATRIX 2 4 2 INDIRECT SOLUTION BY MODAL TRANSFORMATION 2 4 3 GENERAL SOLUTION IN THE PARTICULAR CASE OF HARMONIC EXCITATION 2 5 APPLICATIONS TO MECHANICAL STRUCTURES 2 5 1 DISCRETE MECHANICAL STRUCTURE AT N DEGREES OF FREEDOM, LINEAR, REGULAR AND NON-DISSIPATIVE 2 5 2 DISCRETE MECHANICAL STRUCTURE AT N DOF, LINEAR, REGULAR AND DISSIPATIVE 2 5 3 INTRINSIC VECTOR NORM 2 5 4 PARTICULAR SOLUTION OF THE SYSTEM WITH A HARMONIC FORCE 2 6 INVERSE PROBLEMS: EXPRESSIONS OF THE M, B, K MATRICES ACCORDING TO THE INTRINSIC SOLUTIONS CHAPTER 3 INTRODUCTION TO LINEAR STRUCTURE DYNAMICS 3 1 INTRODUCTION 3 2 PROBLEMS IN STRUCTURE DYNAMICS 3 2 1 FINITE ELEMENTS METHOD 3 2 2 MODAL SUPERPOSITION METHOD 3 2 3 DIRECT INTEGRATION 3 2 4 NEWMARK METHOD 3 2 5 THE ? WILSON METHOD 3 2 6 MODAL ANALYSIS OF THE SANDWICH BEAM CHAPTER 4 INTRODUCTION TO NONLINEAR DYNAMIC ANALYSIS 4 1 INTRODUCTION 4 2 LINEAR SYSTEMS 4 2 1 GENERALITIES 4 2 2 SIMPLE EXAMPLES OF LARGE DISPLACEMENTS 4 2 3 SIMPLE EXAMPLE OF A VARIABLE 4 2 4 SIMPLE EXAMPLE OF DRY FRICTION 4 2 5 MATERIAL NONLINEARITIES 4 3 THE NONLINEAR 1 DOF SYSTEM 4 3 1 GENERALITIES 4 3 2 MOVEMENT WITHOUT NON-DAMPENED EXCITATION 4 3 3 CASE OF A STIFFNESS IN THE FORM ? (1 + ??2) 4 3 4 MOVEMENT WITH NON-DAMPENED EXCITATION 4 3 5 MOVEMENT WITH DAMPENED EXCITATION 4 4 NONLINEAR N DOF SYSTEMS 4 4 1 GENERALITIES 4 4 2 NONLINEAR CONNECTION WITH PERIODIC MOVEMENT 4 4 3 DIRECT INTEGRATION OF THE EQUATIONS CHAPTER 5 CONDENSATION METHODS APPLIED TO EIGEN VALUE PROBLEMS 5 1 INTRODUCTION 5 2 MATHEMATICAL GENERALITY: MATRIX TRANSFORMATION 5 3 DYNAMIC CONDENSATION METHODS 5 4 GUYAN CONDENSATION 5 5 RAYLEIGH–RITZ METHOD 5 6 CASE OF A TEMPORARY PROBLEM 5 6 1 SIMPLIFICATION WITH A FULL MODAL BASIS CHAPTER 6 LINEAR SUBSTRUCTURE APPROACH FOR DYNAMIC ANALYSIS 6 1 GENERALITIES 6 2 DIFFERENT TYPES OF RITZ VECTORS 6 2 1 STRESS VECTORS OF THE JST SUBSTRUCTURE ??(J) 6 2 2 ATTACHMENT VECTORS OF THE JST SUBSTRUCTURE ??(J) 6 2 3 DISPLACEMENT FIELD TYPE VECTORS IN DYNAMIC REGIMES 6 3 SYNTHESIS OF EIGEN SOLUTIONS OF THE ASSEMBLED STRUCTURE: FORMULATION BY AN ENERGETIC METHOD (LAGRANGE WITH MULTIPLICATORS) 6 3 1 EQUILIBRIUM EQUATION OF THE KST ISOLATED SUBSTRUCTURE ??(K) 6 3 2 RITZ BASIS FOR THE KTHE SUBSTRUCTURE ??(K) 6 3 3 COMPATIBILITIES BETWEEN SUBSTRUCTURE ??(1) AND ??(2) 6 3 4 LAGRANGIAN L OF THE ASSEMBLED STRUCTURE 6 4 CRAIG AND BAMPTON SUBSTRUCTURATION METHOD 6 4 1 FORMULATION OF BASE RELATIONS IN THE CASE OF TWO SUBSTRUCTURES 6 4 2 ASSEMBLY OF TWO SUBSTRUCTURES 6 4 3 RESTORING PHYSICAL DOF 6 4 4 COMMENTS 6 5 MIXED METHOD 6 5 1 FORMATION IN THE CASE OF A SINGLE SECONDARY SS 6 5 2 RECONSTRUCTING THE ASSEMBLED STRUCTURE 6 5 3 COMMENTS 6 6 METHODS WITH EIGEN VECTORS WITH FREE COMMON CONTOURS 6 6 1 STIFFNESS METHOD OF COUPLING 6 6 2 SOLUTION TO [6 39] RITZ TRANSFORMATION 6 6 3 FORMULATION BASED ON THE DYNAMIC FLEXIBILITY MATRICES: SEARCH FOR THE ASSEMBLED STRUCTURE’S EIGEN SOLUTIONS 6 6 4 FORMULATION IN THE CASE OF TWO ??(K), K = 1,2, ETC 6 7 METHOD SYSTEMATICALLY INTRODUCING AN INTERMEDIARY CONNECTION STRUCTURE 6 7 1 FORMATION 6 7 2 INTRODUCING RITZ VECTORS 6 7 3 INTRODUCING FITTING CONDITIONS 6 7 4 EQUILIBRIUM EQUATIONS OF THE ASSEMBLED STRUCTURE 6 7 5 NORMALIZATION OF THE ASSEMBLED STRUCTURE’S EIGEN VECTORS 6 7 6 CRITIQUE OF THE METHOD CHAPTER 7 NONLINEAR SUBSTRUCTURE APPROACH FOR DYNAMIC ANALYSIS 7 1 INTRODUCTION 7 2 DYNAMIC SUBSTRUCTURATION APPROACHES 7 2 1 LINEAR CASE 7 2 2 NONLINEAR CASE 7 3 NONLINEAR SUBSTRUCTURE APPROACH 7 3 1 VIBRATION EQUATIONS OF A SUBSTRUCTURE 7 3 2 FIXED INTERFACE PROBLEM 7 3 3 STATIC RAISING PROBLEM 7 3 4 REPRESENTATION OF THE SYSTEM IN CRAIG-BAMPTON’S LINEAR BASE 7 3 5 MODEL REDUCTION WITH THE SHAW AND PIERRE APPROACH 7 3 6 ASSEMBLING SUBSTRUCTURES 7 4 PROPER ORTHOGONAL DECOMPOSITION FOR FLOWS 7 4 1 PROPERTIES OF THE POD MODES 7 4 2 POD SNAPSHOT 7 4 3 SCRIPT OF LOW-ORDER DYNAMIC SYSTEMS 7 5 NUMERICAL RESULTS 7 5 1 MODAL ANALYSIS 7 5 2 DECOMPOSITION OF THE CIRCULAR ACOUSTIC CAVITY 7 5 3 DECOMPOSITION OF THE ELASTIC RING CHAPTER 8 DIRECT AND INVERSE SENSITIVITY 8 1 INTRODUCTION 8 2 DIRECT SENSITIVITY 8 2 1 DEFINITION OF THE STATE’S SENSITIVITY MATRIX X(T) 8 2 2 SENSITIVITY EQUATIONS 8 2 3 SIMPLE DIRECT APPLICATIONS 8 3 SENSITIVITY OF EIGEN SOLUTIONS 8 3 1 DIRECT NUMERICAL METHOD 8 3 2 DERIVATIVES OF THE EIGEN VECTORS ACCORDING TO THE MODAL BASES 8 3 3 DERIVATIVES OF EIGEN VECTORS BASED ON THE EXACT EXPRESSIONS 8 4 FIRST DERIVATIVE OF A PARTICULAR SOLUTION 8 4 1 SCALAR CASE (PRIMARILY DIDACTIC) 8 4 2 GENERAL CASE 8 5 GROUPING THE SENSITIVITY RELATIONS TOGETHER 8 5 1 VARIATIONS 8 5 2 GROUPING THE EIGEN VALUES AND EIGEN VECTORS TOGETHER 8 6 INVERSE SENSITIVITY 8 6 1 OVERDETERMINED CASE: 2A > M 8 6 2 UNIQUE SOLUTION: 2A = M 8 6 3 UNDERDETERMINED CASE: 2A < M CHAPTER 9 PARAMETRIC IDENTIFICATION AND MODEL ADJUSTMENT IN LINEAR ELASTIC DYNAMICS 9 1 INTRODUCTION 9 2 STUDY IN THE ELASTIC DYNAMICS OF MECHANICAL STRUCTURES 9 2 1 PROVISIONAL CALCULATIONS OF BEHAVIOR BASED ON MATHEMATICAL MODELS 9 2 2 IDENTIFICATION 9 3 PARAMETRIC IDENTIFICATION – USE OF A TEST FOR CONSTRUCTING WEAKER CALCULATION MODELS 9 3 1 INTRODUCTION 9 3 2 ERROR MINIMIZATION IN THE BEHAVIORAL EQUATION 9 3 3 ERROR MINIMIZATION ON THE OUTPUTS 9 3 4 COMBINED ESTIMATION OF THE STATE AND THE PARAMETERS 9 4 SOME BASIC METHODS IN PARAMETRIC IDENTIFICATION 9 4 1 LINEAR DEPENDENCY WITH RESPECT TO THE PARAMETERS AND ESTIMATION IN THE SENSE OF THE LEAST SQUARES 9 4 2 ESTIMATION OF PARAMETERS IN THE SENSE OF MAXIMUM LIKELIHOOD 9 4 3 ESTIMATION OF THE VECTOR P BY THE GAUSS–NEWTON METHOD BAYES FORMULATION VECTOR Z(P) NONLINEAR FUNCTION OF P 9 4 4 NON-RANDOM LEAST SQUARES METHOD 9 4 5 QUASI-LINEARIZATION METHOD 9 5 PARAMETRIC CORRECTION OF FINITE ELEMENTS MODELS IN LINEAR ELASTIC DYNAMICS BASED ON THE TEST RESULTS 9 5 1 HIGHLIGHTING A FEW DIFFICULTIES 9 6 M MODEL ADJUSTMENT: K??C, C BY MINIMIZING THE MATRIX NORMS BY THE CORRECTION MATRICES ?M, ?K 9 6 1 PRINCIPLE OF BARUCH AND BAR-ITZHACK METHOD 9 6 2 KABE, SMITH AND BEATTIE METHODS 9 7 M MODEL ADJUSTMENT: K??C, C BY MINIMIZING RESIDUE VECTORS MADE UP BASED ON LOCAL CORRECTION MATRICES ?MI , ?KI 9 7 1 MINIMIZATION OF FORMED RESIDUE BASED ON THE BEHAVIOR EQUATION 9 7 2 MINIMIZATION OF FORMED RESIDE BASED ON OUTPUTS CHAPTER 10 INVERSE PROBLEMS IN DYNAMICS: ROBUSTNESS FUNCTION 10 1 INTRODUCTION 10 2 CONVEX MODELS 10 2 1 DEFINITIONS 10 2 2 DIRECT PROBLEM 10 2 3 INVERSE PROBLEM 10 3 ROBUSTNESS FUNCTION 10 3 1 MONOCRITERION RESPONSE 10 3 2 MULTICRITERIA RESPONSE 10 4 SOLUTION METHODS 10 4 1 INTERVAL ARITHMETIC 10 4 2 OPTIMIZATION METHOD 10 5 NUMERICAL CALCULATIONS 10 6 APPLICATIONS 10 6 1 DUAL-RECESSED BEAM 10 6 2 SQUARE 10 7 CONCLUSION CHAPTER 11 MODAL SYNTHESIS AND RELIABILITY OPTIMIZATION METHODS 11 1 INTRODUCTION 11 2 DESIGN RELIABILITY OPTIMIZATION IN STRUCTURAL DYNAMICS 11 2 1 FREQUENTIAL HYBRID METHOD 11 2 2 OPTIMIZATION CONDITION OF THE HYBRID PROBLEM 11 3 THE SP METHOD 11 3 1 FORMULATION OF THE PROBLEM 11 3 2 IMPLEMENTATION OF THE SP APPROACH 11 4 MODAL SYNTHESIS AND RBDO COUPLING METHODS 11 5 DISCUSSION